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Hyperloop Homework

The hyperloop is an idea proposed by Elon Musk. The basic idea is to build a system of tubes with passenger carrying pods inside. The tubes would have reduced air pressure and pods would have a fan to both make it hover and to push it forward. With this, Elon Musk says that you can get from L.A. to San Francisco in just 30 minutes. You can read all the awesome details at this hyperloop page (with a pdf).

Remember, these are just questions. I don’t always give all the data needed to answer the questions and it’s possible a question could have no valid answer. You might have to make some estimates or look up some information to answer these. It would be safe to assume that you also might need some numerical models to answer some questions. Some of these questions have answers in the hyperloop-alpha document. However, you should still find the answer yourself – you know, just to check. Let’s begin.

Air Resistance

The claim is that a high speed railway just won’t work. The problem is with the air. Suppose you have two high speed trains. Train 1 is your run of the mill high speed train (with a speed around 100 – 300 mph). Train 2 is EXACTLY the same except that it is in a region with half the density of air. Train 3 is just like the hyperloop with an air pressure of 100 Pascals. Make a power vs. speed plot for all three of these trains. How fast would train 2 go if it used the same power as the normal air high speed train? What about train 3? If you like, you can use the following model for magnitude of the air resistance force:

In this model, ρ is the density of air. A is the cross sectional area of the train and C is a drag coefficient that depends on shape. If you like, you can ignore the frictional effects with the ground.

In the alpha paper, the hyperloop pod is listed as requiring 100 kWatts at 700 mph (313 m/s) with a drag force of 320 Newtons. Estimate the product of AC for this pod. Yes, I know that the aerodynamics inside a tube are not the same as plain air.

The pod is estimated to have a mass of 3100 kg. If it is moving at an initial speed of 313 m/s, how long would it take to get to a speed of 10 m/s with just air resistance.

Capacity and Cost

Let’s say that it takes 30 minutes to go from LA to SF and each hyperloop pod carries a maximum of 28 passengers – but maybe an average of 20 passengers per pod. If a pod leaves every 30 minutes, how long would it take to move all the population of LA to San Francisco?

If the hyperloop costs 6 billion USD to build, how quick could this investment be earned back with a $20 one-way ticket? Of course, this assumes that the hyperloop is energy neutral (it generates its own energy with solar panels). How does the $20 one-way ticket compare to the price it would cost to drive from LA to SF?

The hyperloop alpha document has a graph showing the energy cost per passenger for different transportation modes. Based on my estimation, it gives the following for a trip from LA to SF.

Car: 811 MJ per passenger.

Motorcycle: 975 MJ per passenger.

Airplane: 1025 MJ per passenger.

Train: 861 MJ per passenger.

Model S: 285 MJ per passenger.

Passenger Hyperloop: 62 MJ per passenger.

Passenger + Vehicle Hyperloop: 248 MJ per passenger.

Ignore the last item. I think it refers to a hyperloop pod that carries passengers as well as 3 cars. However, for the other items see if you can “fact check” these numbers. Make an estimate and see if these values are in the right ballpark for the energy required per passenger for the trip.

If the internal battery has a mass of 2,500 kg – what is the energy density for this battery? What type of battery could it be?

Each pod requires 100 kW. How wide would the solar panels on top of the tubes need to be in order to provide enough energy to run the hyperloop for 24 hours. Don’t forget – the Sun doesn’t shine during the night.

Acceleration

The hyperloop will use linear induction motors for accelerations. I’m not completely certain, but I interpret this to mean that there will be sections of the tube with the accelerating elements. If the pod has an acceleration of 9.8 m/s2, how long would the accelerating element be for a speed change from 0 to 300 mph (134 m/s)? How long would an acceleration element be to change the speed from 300 mph to 700 mph with the same acceleration?

If the pod accelerates horizontally with a magnitude of 9.8 m/s2, how many total “g’s” would the passenger feel? Remember, if you just stand there you feel 1 g.

The document states that the maximum perpendicular acceleration due to a turn will be 0.5 gs. Make a plot of radius of turning vs. speed such that acceleration is a constant 0.5 g.

Here is a view of a round-a-bout near my house. From Google maps, it appears to have a diameter of about 35 meters.

Screen capture from Google Maps

While driving around this circle, a speed of 15 mph feels very reasonable. What is the lateral (perpendicular) acceleration I experience during this turn? How does that compare to the hyperloop?

The Hyperloop alpha document lists the following for the radius of curvature of the tube at different speeds.

Radius of 3.67 km at 300 mph.

Radius of 12.6 km at 555 mph.

Radius of 23.5 km at 760 mph.

Does this give a lateral acceleration of less than .5 g?

Kinematics

The Hyperloop Alpha document shows a graph of speed vs. time for the pod. I used Tracker Video to digitize this data and convert it to m/s instead of mph.

I also have all the data points used to create this graph in a Google Docs spreadsheet. Use this graph (or the spreadsheet data) to produce both a plot of acceleration vs. time and position vs. time. What is the maximum acceleration in this motion?

Ok, that’s it for now. If you can answer all these questions, you might be the next Elon Musk (or Tony Stark).